The Penrose and Hawking Singularity Theorems Revisited

Home , Causal structure, Energy condition

Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook

The Penrose and Hawking Singularity Theorems revisited

Roland Steinbauer

Faculty of Mathematics, University of Vienna

Prague, October 2016

The Penrose and Hawking Singularity Theorems revisited 1 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Overview

Long-term project on Lorentzian geometry and general relativity with metrics of low regularity

jointly with ‘theoretical branch’ (Vienna & U.K.): Melanie Graf, James Grant, GüntherHörmann,Mike Kunzinger, Clemens Sämann,James Vickers ‘exact solutions branch’ (Vienna & Prague): Jiˇr´ıPodolský,Clemens Sämann,Robert Svarcˇ

The Penrose and Hawking Singularity Theorems revisited 2 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Contents

1 The classical singularity theorems

2 Interlude: Low regularity in GR

3 The low regularity singularity theorems

4 Key issues of the proofs

5 Outlook

The Penrose and Hawking Singularity Theorems revisited 3 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Table of Contents

1 The classical singularity theorems

2 Interlude: Low regularity in GR

3 The low regularity singularity theorems

4 Key issues of the proofs

5 Outlook

The Penrose and Hawking Singularity Theorems revisited 4 / 27 Theorem (Pattern singularity theorem [Senovilla, 98]) In a spacetime the following are incompatible (i) Energy condition (iii) Initial or boundary condition (ii) Causality condition (iv) Causal geodesic completeness

(iii) initial condition ; causal geodesics start focussing (i) energy condition ; focussing goes on ; focal point (ii) causality condition ; no focal points way out: one causal geodesic has to be incomplete, i.e., ¬ (iv)

Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Singularities in GR

singularities occur in exact solutions; high degree of symmetries singularities as obstruction to extend causal geodesics [Penrose, 65]

The Penrose and Hawking Singularity Theorems revisited 5 / 27 (iii) initial condition ; causal geodesics start focussing (i) energy condition ; focussing goes on ; focal point (ii) causality condition ; no focal points way out: one causal geodesic has to be incomplete, i.e., ¬ (iv)

Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Singularities in GR

singularities occur in exact solutions; high degree of symmetries singularities as obstruction to extend causal geodesics [Penrose, 65]

Theorem (Pattern singularity theorem [Senovilla, 98]) In a spacetime the following are incompatible (i) Energy condition (iii) Initial or boundary condition (ii) Causality condition (iv) Causal geodesic completeness

The Penrose and Hawking Singularity Theorems revisited 5 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Singularities in GR

singularities occur in exact solutions; high degree of symmetries singularities as obstruction to extend causal geodesics [Penrose, 65]

The Penrose and Hawking Singularity Theorems revisited 5 / 27 Theorem ([Hawking, 1967] Big Bang) A spacetime is future timelike geodesically incomplete, if (i) Ric (X , X ) ≥ 0 for every timelike vector X (ii) There exists a compact space-like hypersurface S in M (iii) The unit normals to S are everywhere converging, θ := −tr K < 0.

Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook The classical theorems Theorem ([Penrose, 1965] Gravitational collapse) A spacetime is future null geodesically incomplete, if (i) Ric (X , X ) ≥ 0 for every null vector X (ii) There exists a non-compact Cauchy hypersurface S in M (iii) There exists a trapped surface (cp. achronal spacelike 2-srf. w. past-pt. timelike mean curvature)

The Penrose and Hawking Singularity Theorems revisited 6 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook The classical theorems Theorem ([Penrose, 1965] Gravitational collapse) A spacetime is future null geodesically incomplete, if (i) Ric (X , X ) ≥ 0 for every null vector X (ii) There exists a non-compact Cauchy hypersurface S in M (iii) There exists a trapped surface (cp. achronal spacelike 2-srf. w. past-pt. timelike mean curvature)

Theorem ([Hawking, 1967] Big Bang) A spacetime is future timelike geodesically incomplete, if (i) Ric (X , X ) ≥ 0 for every timelike vector X (ii) There exists a compact space-like hypersurface S in M (iii) The unit normals to S are everywhere converging, θ := −tr K < 0.

The Penrose and Hawking Singularity Theorems revisited 6 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Hawking’s Thm: proof strategy (C2-case)

Analysis: θ evolves along the normal geodesic congruence of S by Raychaudhury’s equation θ2 θ0 + + Ric(γ, ˙ γ˙ ) + tr(σ2) = 0 3 (i) =⇒ θ0 + (1/3)θ2 ≤ 0 =⇒ (θ−1)0 ≥ 1/3 (iii) =⇒ θ(0) < 0 =⇒ θ → ∞ in ﬁnite time =⇒ focal point Causality theory: ∃ longest curves in the Cauchy development =⇒ no focal points in the Cauchy development + completeness =⇒ D (S) ⊆ exp([0, T ] · nS ). . . compact =⇒ horizon H+(M) compact, ; 2 possibilities (1) H+(M) = ∅. Then I +(S) ⊆ D+(S) =⇒ timlike incomplete (2) H+(M) 6= ∅ compact =⇒ p 7→ d(S, p) has min on H+(S) But from every point in H+(M) there starts a past null generator γ (inextendible past directed null geodesic contained in H+(S)) and p 7→ d(S, p) strictly decreasing along γ =⇒ unbounded

The Penrose and Hawking Singularity Theorems revisited 7 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Regularity for the singularity theorems of GR

Pattern singularity theorem [Senovilla, 98] In a C2-spacetime the following are incompatible (i) Energy condition (iii) Initial or boundary condition (ii) Causality condition (iv) Causal geodesic completeness

Theorem allows (i)–(iv) and g ∈ C1,1 ≡ C2−. But C1,1-spacetimes are physically reasonable models are not really singular (curvature bounded) still allow unique solutions of geodesic eq. ; formulation sensible

Moreover below C1,1 we have unbded curv., non-unique geos, no convexity ; ‘really singular’ Hence C1,1 is the natural regularity class for singularity theorems!

The Penrose and Hawking Singularity Theorems revisited 8 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Table of Contents

1 The classical singularity theorems

2 Interlude: Low regularity in GR

3 The low regularity singularity theorems

4 Key issues of the proofs

5 Outlook

The Penrose and Hawking Singularity Theorems revisited 9 / 27 Why is it needed?

1 Physics: Realistic matter models ; g 6∈ C2 2 Analysis: ivp solved in Sobolev spaces ; g ∈ H5/2(M)

Where is the problem? Physics and Analysis vs. Lorentzian geometry want/need low regularity needs high regularity

But isn’t it just a silly game for mathematicians? NO! Low regularity really changes the geometry!

Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Low regularity GR What is is? GR and Lorentzian geometry on spacetime manifolds (M, g), where M is smoot but g is non-smooth (below C2)

The Penrose and Hawking Singularity Theorems revisited 10 / 27 Where is the problem? Physics and Analysis vs. Lorentzian geometry want/need low regularity needs high regularity

But isn’t it just a silly game for mathematicians? NO! Low regularity really changes the geometry!

Why is it needed?

1 Physics: Realistic matter models ; g 6∈ C2 2 Analysis: ivp solved in Sobolev spaces ; g ∈ H5/2(M)

The Penrose and Hawking Singularity Theorems revisited 10 / 27 But isn’t it just a silly game for mathematicians? NO! Low regularity really changes the geometry!

Why is it needed?

1 Physics: Realistic matter models ; g 6∈ C2 2 Analysis: ivp solved in Sobolev spaces ; g ∈ H5/2(M)

Where is the problem? Physics and Analysis vs. Lorentzian geometry want/need low regularity needs high regularity

The Penrose and Hawking Singularity Theorems revisited 10 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Low regularity GR What is is? GR and Lorentzian geometry on spacetime manifolds (M, g), where M is smoot but g is non-smooth (below C2)

Why is it needed?

1 Physics: Realistic matter models ; g 6∈ C2 2 Analysis: ivp solved in Sobolev spaces ; g ∈ H5/2(M)

Where is the problem? Physics and Analysis vs. Lorentzian geometry want/need low regularity needs high regularity

But isn’t it just a silly game for mathematicians? NO! Low regularity really changes the geometry!

The Penrose and Hawking Singularity Theorems revisited 10 / 27 (2) Initial value problem—Analysis Local existence and uniqueness Thms. for Einstein eqs. in terms of Sobolev spaces classical [CB,HKM]: g ∈ H5/2 =⇒ C1(Σ) recent big improvements [K,R,M,S]: g ∈ C0(Σ)

Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Why Low Regularity?

(1) Realistic matter—Physics

want discontinuous matter conﬁgurations ; T 6∈ C0 =⇒ g 6∈ C2 ﬁnite jumps in T ; g ∈ C1,1 (derivatives locally Lipschitz) more extreme situations (impulsive waves): g piecew. C3, globally C0

The Penrose and Hawking Singularity Theorems revisited 11 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Why Low Regularity?

(1) Realistic matter—Physics

(2) Initial value problem—Analysis Local existence and uniqueness Thms. for Einstein eqs. in terms of Sobolev spaces classical [CB,HKM]: g ∈ H5/2 =⇒ C1(Σ) recent big improvements [K,R,M,S]: g ∈ C0(Σ)

The Penrose and Hawking Singularity Theorems revisited 11 / 27 the y-axis is a geodesic which is non-minimising between any of its points

BUT shortest curves from (0,0) to (0,y) are two symmetric arcs ; minimising curves not unique, even locally

Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Low regularity changes the geometry

Riemannian counterexample[ Hartman&Wintner, 51] 1 0 g (x, y) = on (−1, 1) × ⊆ 2 ij 0 1 − |x|λ R R

λ ∈ (1, 2) =⇒ g ∈ C1,λ−1 H¨older,slightly below C1,1 (nevertheless) geodesic equation uniquely solvable

The Penrose and Hawking Singularity Theorems revisited 12 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Low regularity changes the geometry

Riemannian counterexample[ Hartman&Wintner, 51] 1 0 g (x, y) = on (−1, 1) × ⊆ 2 ij 0 1 − |x|λ R R

λ ∈ (1, 2) =⇒ g ∈ C1,λ−1 H¨older,slightly below C1,1 (nevertheless) geodesic equation uniquely solvable

BUT shortest curves from (0,0) to (0,y) are two symmetric arcs ; minimising curves not unique, even locally

the y-axis is a geodesic which is non-minimising between any of its points

The Penrose and Hawking Singularity Theorems revisited 12 / 27 Lorentzian geometry and regularity classically g ∈ C∞, for all practical purposes g ∈ C2 things go wrong below C2 convexity goes wrong for g ∈ C1,α (α < 1) [HW, 51] causality goes wrong, light cones “bubble up” for g ∈ C0 [CG, 12]

Things that can be done

impulisve grav. waves g ∈ Lip, D0 [J.P., R.S.,ˇ C.S., R.S., A.L.] causality theory for continuous metrics [CG, 12], [S¨amann,16] singularity theorems in C1,1 [KSSV, 15], [KSV, 15]

Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook GR and low regularity The challenge Physics and Analysis vs. Lorentzian geometry want/need low regularity needs high regularity

The Penrose and Hawking Singularity Theorems revisited 13 / 27 Things that can be done

impulisve grav. waves g ∈ Lip, D0 [J.P., R.S.,ˇ C.S., R.S., A.L.] causality theory for continuous metrics [CG, 12], [S¨amann,16] singularity theorems in C1,1 [KSSV, 15], [KSV, 15]

Lorentzian geometry and regularity classically g ∈ C∞, for all practical purposes g ∈ C2 things go wrong below C2 convexity goes wrong for g ∈ C1,α (α < 1) [HW, 51] causality goes wrong, light cones “bubble up” for g ∈ C0 [CG, 12]

The Penrose and Hawking Singularity Theorems revisited 13 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook GR and low regularity The challenge Physics and Analysis vs. Lorentzian geometry want/need low regularity needs high regularity

Things that can be done

impulisve grav. waves g ∈ Lip, D0 [J.P., R.S.,ˇ C.S., R.S., A.L.] causality theory for continuous metrics [CG, 12], [S¨amann,16] singularity theorems in C1,1 [KSSV, 15], [KSV, 15]

The Penrose and Hawking Singularity Theorems revisited 13 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Table of Contents

1 The classical singularity theorems

2 Interlude: Low regularity in GR

3 The low regularity singularity theorems

4 Key issues of the proofs

5 Outlook

The Penrose and Hawking Singularity Theorems revisited 14 / 27 Theorem allows (i)–(iv) and g ∈ C1,1. But C1,1-spacetimes are physically okay/not singular allow to formulate the theorems

C1,1 is the natural regularity class for the singularity theorems.

Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Again: Why go to C1,1?

Recall: Theorem (Pattern singularity theorem [Senovilla, 98]) In a C2-spacetime the following are incompatible (i) Energy condition (iii) Initial or boundary condition (ii) Causality condition (iv) Causal geodesic completeness

The Penrose and Hawking Singularity Theorems revisited 15 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Again: Why go to C1,1?

Theorem allows (i)–(iv) and g ∈ C1,1. But C1,1-spacetimes are physically okay/not singular allow to formulate the theorems

C1,1 is the natural regularity class for the singularity theorems.

The Penrose and Hawking Singularity Theorems revisited 15 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook The classical Theorems Theorem [Hawking, 1967] A C2 -spacetime is future timelike geodesically incomplete, if (i) Ric (X , X ) ≥ 0 for every timelike vector X (ii) There exists a compact space-like hypersurface S in M (iii) The unit normals to S are everywhere converging

Theorem [Penrose, 1965] A C2 -spacetime is future null geodesically incomplete, if (i) Ric (X , X ) ≥ 0 for every null vector X (ii) There exists a non-compact Cauchy hypersurface S in M (iii) There exists a trapped surface T (cp. achronal spacelike 2-srf. w. past-pt. timelike mean curvature)

The Penrose and Hawking Singularity Theorems revisited 16 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook The C1,1-Theorems Theorem [Kunzinger, S., Stojkovi´c,Vickers, 2015] A C1,1-spacetime is future timelike geodesically incomplete, if (i) Ric (X , X ) ≥ 0 for every smooth timelike local vector ﬁeld X (ii) There exists a compact space-like hypersurface S in M (iii) The unit normals to S are everywhere converging

Theorem [Kunzinger, S., Vickers, 2015] A C1,1-spacetime is future null geodesically incomplete, if (i) Ric (X , X ) ≥ 0 for every Lip-cont. local null vector ﬁeld X (ii) There exists a non-compact Cauchy hypersurface S in M (iii) There exists a trapped surface T (cp. achronal spacelike 2-srf. w. past-pt. timelike mean curvature)

The Penrose and Hawking Singularity Theorems revisited 16 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Obstacles in the C1,1-case

No appropriate version of calculus of variations available (second variation, maximizing curves, focal points, index form, . . . ) C2-causality theory rests on local equivalence with Minkowski space. This requires good properties of exponential map. ; big parts of causality theory have to be redone Ricci tensors is only L∞ ; problems with energy conditions

strategy: Proof that the exponential map is a bi-Lipschitz homeo Re-build causality theory for C1,1-metrics regularisation adapted to causal structure replacing calculus of var. use surrogate energy condition

The Penrose and Hawking Singularity Theorems revisited 17 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Table of Contents

1 The classical singularity theorems

2 Interlude: Low regularity in GR

3 The low regularity singularity theorems

4 Key issues of the proofs

5 Outlook

The Penrose and Hawking Singularity Theorems revisited 18 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook The exponential map in low regularity

expp : TpM 3 v 7→ γv (1) ∈ M, where γv is the (unique) geodesic starting at p in direction of v 2 g ∈ C ⇒ expp local diﬀeo 1,1 g ∈ C ⇒ expp loc. homeo [W,32]

Optimal regularity 1,1 g ∈ C ⇒ expp bi-Lipschitz homeo [KSS,14]: regularisation & comparison geometry [Minguzzi,15]: reﬁned ODE methods

; bulk of causality theory remains true in C1,1 [CG,12, KSSV,14, Ming.,15]

The Penrose and Hawking Singularity Theorems revisited 19 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Chrusciel-Grant regularization of the metric

Regularisation adapted to the causal structure [CG,12], [KSSV, 14]

Sandwich null cones of g between null cones of two approximating families of smooth metrics so that

gˇε ≺ g ≺ gˆε.

applies to continuous metrics local convolution plus small shift

Properties of the approximations for g ∈ C1,1

1 (i) gˇε, gˆε → g locally in C 2 2 (ii) D gˇε, D gˆε locally uniformly bded. in ε, but Ric[gε] 6→ Ric[g]

The Penrose and Hawking Singularity Theorems revisited 20 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Surrogate energy condition (Hawking case)

Lemma [KSSV, 15] Let (M, g) be a C1,1-spacetime satisfying the energy condition Ric [g](X , X ) ≥ 0 for all timelike local C∞-vector ﬁelds X .

Then for all K ⊂⊂ M ∀C > 0 ∀δ > 0 ∀κ < 0 ∀ε small

Ric [gˇε](X , X ) > −δ ∀X ∈TM|K : gˇε(X , X ) ≤ κ, kX kh ≤ C.

2 Proof. gˇε − g ∗ ρε → 0 in C ; only consider gε := g ∗ ρε i i i i m i m Rjk = Rjki = ∂xi Γkj − ∂xk Γij +Γ imΓkj − ΓkmΓij Blue terms|ε converge uniformly For red terms use variant of Friedrich’s Lemma: ρε ≥ 0 =⇒ Ric[g](X , X ) ∗ ρε ≥ 0 Ric[g](X , X ) ∗ ρε − Ric[gε](X , X ) → 0 unif. The Penrose and Hawking Singularity Theorems revisited 21 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook The C1,1-proof (Hawking case)

D+(S) ⊆ D+ (S): gˇε

+ D (S) gε

p D+ (S)

J - (p) S

Limiting argument ⇒ ∃ maximising g-geodesic γ for all p ∈ D+(S) 1 and γ = lim γgˇε in C

Surrogate energy condition for gˇε and Raychaudhury equation ⇒ D+(S) relatively compact otherwise ∃ gˇε-focal pt. too early ⇒ H+(S) ⊆ D+(S) compact Derive a contradiction as in the C∞-case using C1,1-causality

The Penrose and Hawking Singularity Theorems revisited 22 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Surrogate energy condition (Penrose case)

Lemma [KSV, 15] Let (M, g) be a C1,1-spacetime satisfying the energy condition Ric [g](X , X ) ≥ 0 for every local Lip. null vector ﬁeld X .

Then for all K ⊂⊂ M ∀C > 0 ∀δ > 0 ∃η > 0 s.t. we have

Ric [gˆε](X , X ) > −δ

for all p ∈ K and all X ∈ TpM with kX kh ≤ C which are close to a g-null vector in the sense that

∃Y0 ∈ TM|K g-null, kY0kh ≤ C, dh(X , Y0) ≤ η.

The Penrose and Hawking Singularity Theorems revisited 23 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook The C1,1-proof (Penrose case)

Choose gˆε globally hyperbolic (stability [NM,11], [S,15]) Surrogate energy condition is strong enough to guarantee that

+ + + Eε (T ) = Jε (T ) \ Iε (T ) is relatively compact

in case of null geodesic completeness

gˆε globally hyperbolic ⇒

+ + 0 Eε (T ) = ∂Jε (T ) is a gˆε-achronal, compact C -hypersrf.

+ g < gˆε ⇒ Eε (T ) is g-achronal derive usual (topological) contradiction

The Penrose and Hawking Singularity Theorems revisited 24 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Table of Contents

1 The classical singularity theorems

2 Interlude: Low regularity in GR

3 The low regularity singularity theorems

4 Key issues of the proofs

5 Outlook

The Penrose and Hawking Singularity Theorems revisited 25 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook

Lemma [Hawking and Penrose, 1967] In a causally complete C2-spacetime, the following cannot all hold: 1 Every inextendible causal geodesic has a pair of conjugate points 2 M contains no closed timelike curves and 3 there is a future or past trapped achronal set S

Theorem A C2-spacetime M is causally incomplete if Einstein’s eqs. hold and

1 M contains no closed timelike curves 2 M satisﬁes an energy condition 3 Genericity: nontrivial curvature at some pt. of any causal geodesic 4 M contains either a trapped surface some p s.t. convergence of all null geodesics changes sign in the past a compact spacelike hypersurface

The Penrose and Hawking Singularity Theorems revisited 26 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook

Theorem A C2-spacetime M is causally incomplete if Einstein’s eqs. hold and

The Penrose and Hawking Singularity Theorems revisited 26 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Some related Literature

[CG,12] P.T. Chrusćiel,J.D.E. Grant, On Lorentzian causality with continuous metrics. CQG 29 (2012) [KSS,14] M. Kunzinger, R. Steinbauer, M. Stojković, The exponential map of a C1,1-metric. Diff. Geo. Appl.34(2014) [KSSV,14] M. Kunzinger, R. Steinbauer, M. Stojković,J.A. Vickers, A regularisation approach to causality theory for C1,1-Lorentzian metrics. GRG 46 (2014) [KSSV,15] M. Kunzinger, R. Steinbauer, M. Stojković,J.A. Vickers, Hawking’s singularity theorem for C 1,1-metrics. CQG 32 (2015) [KSV,15] M. Kunzinger, R. Steinbauer, J.A. Vickers, The Penrose singularity theorem in C 1,1. CQG 32 (2015) [LSS,14]ˇ A. Lecke, R. Steinbauer, R. Svarc,ˇ The regularity of geodesics in impulsive pp-waves. GRG 46 (2014) [PSS,14] J. Podolský,R. Steinbauer, R. Svarc,ˇ Gyratonic pp-waves and their impulsive limit. PRD 90 (2014) [PSSS,15]ˇ J. Podolský,C. Sämann,R. Steinbauer, R. Svarc,ˇ The global existence, uniqueness and C1-regularity of geodesics in nonexpanding impulsive gravitational waves. CQG 32 (2015) [PSSS,16]ˇ J. Podolský,C. Sämann,R. Steinbauer, R. Svarc,ˇ The global uniqueness and C 1-regularity of geodesics in expanding impulsive gravitational waves. arXiv:1602.05020, to appear in CQG [SS,12] C. Sämann,R. Steinbauer, On the completeness of impulsive gravitational wave spacetimes. CQG 29 (2012) [SS,15] C. Sämann,R. Steinbauer, Geodesic completeness of generalized space-times. in Pseudo-differential operators and generalized functions. Pilipovic, S., Toft, J. (eds) Birkhäuser/Springer, 2015 [SSLP,16] C. Sämann,R. Steinbauer, A. Lecke, J. Podolský, Geodesics in nonexpanding impulsive gravitational waves with Λ, part I, CQG 33 (2016) [SSS,16]ˇ C. Sämann,R. Steinbauer, R. Svarc,ˇ Completeness of general pp-wave spacetimes and their impulsive limit. arXiv:1607.01934, to appear in CQG [S,14] R. Steinbauer, Every Lipschitz metric has C 1-geodesics. CQG 31, 057001 (2014) Thank you for your attention!

The Penrose and Hawking Singularity Theorems revisited 27 / 27